Clans with zero on an interval

Abstract

Following the terminology of Wallace [8] we shall use the word mob to mean a Hausdorff topological semigroup, and shall use clan for a compact connected mob with unit. Interval means a closed interval on the real line, although as A. H. Clifford has pointed out to the authors, nearly all the theorems (and proofs) generalize to arbitrary compact connected linearly ordered topological spaces.The object of this paper is to characterize clans with zero on an interval. Partial results in this connection have been found by Faucett [3; 4] and Clifford [l]. In addition the case when 0 (the zero) is an end point has been studied by Mostert and Shields [5]. Finally a forthcoming paper of Clifford In what follows 5 is always a clan on an interval with zero. It is well known (e.g. Wallace [7]) that the unit u is an end point. We will assume that it is the right hand end point (the other case, of course, can be handled by a dual argument) and call the other end point 5. Let L be the interval [S, 0] and R the interval [0, u] so that we have the following diagram for S: L R S 0 u Fig. 1 We define a partial order ■< on 5 as follows: x<y if and only if x separates y and 0, (i.e. x <y if both x and y are on the same side of 0 and x is closer to 0 than y is). We use the notation /, hEL and r, riERThe authors would like to express their gratitude to Professor R. J. Koch for suggesting the question considered here as well as for patient listening and helpful suggestions.1. We look first at the case when L is degenerate. In this case 8 = 0 and 0 is an endpoint. These clans have been completely determined by the work of Mostert and Shields, and Clifford as noted above. For completeness we include a summary of their results in this section.Let S be such a clan and let E = {s| sES and s2 = s} (i.e. the idempotents

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“…Let S be a globally idempotent thread having a zero which is a cut point. We show first that, by passing to one of the duals of S if necessary, we can achieve a thread which satisfies (4) and, of course, (1) and 2, and in which G2 = G.…”

“…[February is a positive thread provided the zero is a least element and provided there is no largest element. The structure theorem for standard threads can be found in [1], [4], [5] or [6]; any positive thread is the contact extension (see §1) of a standard thread by the thread of nonnegative real numbers under ordinary multiplication. In a thread with a zero, we write x^yif and only if 0^x<y or y<x = 0.…”

“…We shall say that a thread 5 is the contact extension of T by Q if and only if T and Q are threads satisfying the hypotheses of 1.1 and S is isomorphic with the thread constructed there. The adjective "contact" was used by Clifford in [1] for a similar extension, however, 1.1 and its converse, 1.2, more closely resemble Lemma 6.1 in [4].…”

“…Now suppose that Tis a thread satisfying (1), (2) and (4) in which G2 = G. Then G, being itself a globally idempotent thread and having a zero as a least element, must be a positive thread, or a standard thread, or a standard thread without its identity. In the third case, which is the only one in which Tdoes not already satisfy (3), if we consider only G it is clear that the missing identity can be replaced.…”

Compact Connected Ordered Semitopological Semigroups†

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Compact, connected, totally-ordered semigroups have been extensively investigated under the hypothesis that the multiplication be jointly continuous [2,3,5,6,7,9]. An excellent exposition of this theory may be found in .

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